The Euler characteristic of a category

By Tom Leinster. Documenta Mathematica, 2006

158 citations recorded by the source corpus.

The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula is proved for the cardinality of a colimit of sets, generalizing the classical inclusion-exclusion formula. Both rest on a generalization of Rota's Mobius inversion from posets to categories.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/993bbe9ffd2cd9966e31a13c64f28141b95640e4

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces (2013, methodology): “In [6], Leinster had defined the Euler characteristic of a finite category, which generalizes the Euler characteristic of a topological space or of a poset.”
Categorifying the magnitude of a graph (2015, methodology): “Overview The magnitude of a finite metric space was introduced by Leinster [9] by analogy with his notion of the Euler characteristic of a category [8].”
Maximizing diversity in biology and beyond (2015, unknown): “For instance, it automatically produces a notion of the Euler characteristic of an (ordinary) category, closely related to the topological Euler characteristic [14]; in the context of metric spaces, magnitude encodes geometric information such as volume and dimension [15–17]; in graph theory, magnit”
Magnitude homology of enriched categories and metric spaces (2017, methodology): “e magnitude of X as Mag(X), or Magk(X) or Mag#(X) if necessary to indicate the relevant semiring or size function. Example 2.7. If V = FinSet and # is the cardinality valued in Q, then it is shown in [Lei08] that if X is a ﬁnite ordinary category that is skeletal and contains no nonidentity endomorp”
Higher Dimensional Algebra VII: Groupoidification (2009, unknown): “This in turn provides a new way to categorify— or more precisely, groupoidify—the positive part of the quantum group associated to the quiver.”
The magnitude of metric spaces (2010, methodology): “When V = FinSet, we obtain a notion of the magnitude |A| ∈ Q of a finite category A [17, 3].”
On the asymptotic magnitude of subsets of Euclidean space (2009, background): “In the first part of this section a metric space is viewed as an enriched category so that the magnitude can be defined analogously to the Euler characteristic of a finite category which was introduced in [7] and [1].”
The magnitude of a graph (2014, background): “ely related to Euler characteristic of topological spaces, of more exotic spaces such as orbifolds, and of posets; indeed, its theory generalizes the theory of Mobius{Rota inversion for posets. (See [3], especially Propositions 2.11 and 2.12 and Section 4.) Specialized to metric spaces, dierent conn”